Precalculus Examples

$\log_{2} (\sin (\frac{π}{4}))$

Step 1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\log_{2} (\frac{\sqrt{2}}{2})$

Step 2

Rewrite $\frac{\sqrt{2}}{2}$ as $\sqrt{2} \cdot 2^{- 1}$ .

$\log_{2} (\sqrt{2} \cdot 2^{- 1})$

Step 3

Rewrite $\log_{2} (\sqrt{2} \cdot 2^{- 1})$ as $\log_{2} (\sqrt{2}) + \log_{2} (2^{- 1})$ .

$\log_{2} (\sqrt{2}) + \log_{2} (2^{- 1})$

Step 4

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{2} (\sqrt{2}) - \log_{2} (2)$

Step 5

Logarithm base $2$ of $2$ is $1$ .

$\log_{2} (\sqrt{2}) - 1 \cdot 1$

Step 6

Multiply $- 1$ by $1$ .

$\log_{2} (\sqrt{2}) - 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (\sqrt{2}) - 1$

Decimal Form:

$- 0.5$